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In the census of 1790, the first, assistant marshals in federal judicial districts gathered data from households on all that belonged to them—adult, child, slave, and freeborn. From left, Ellen Knecht, Eomer Knecht, Sylvia Lee, Ben Knecht, and Steve Holloway.

Colonial Williamsburg

Partly on the advice of Secretary of State Thomas Jefferson, who figured things differently, President George Washington vetoed the first attempt at apportionment. Interpreters Bill Barker, left, as Jefferson, and Ron Carnegie as Washington run the numbers.

Labour Society, London

Tom Paine, here in an 1800s portrait by A. Easton, railed against the unequal representation of England’s electoral system.

“The Greatest Practical Approach to Exactness.”

The Problem of Apportionment and Washington's First Veto

by Jack Lynch

It sounds so simple. “Representatives . . .
shall be apportioned among the several States
which may be included within this Union, according
to their respective Numbers.” So reads Article
1, Section 2, Clause 3, of the Constitution of the
United States. Yet this apparently uncomplicated
provision not only led to the nation’s first veto—it
poses a problem that has exercised historians, political
scientists, legal scholars, and mathematicians
for more than two centuries.

The principle is straightforward: more
residents should mean more representatives.
Americans took pride in proportional representation,
contrasting it
with the “rotten boroughs”
in England. Although members of Parliament had been
allotted to the districts more or less according to their
population, centuries of demographic change had been ignored, leading to absurdities.
Once bustling districts, now deserted, retained their
medieval numbers of MPs; burgeoning cities were
stuck with the representation they had half a millennium
before. As Tom Paine wrote in The Rights of Man,
“Old Sarum, which contains
not three houses, sends two
members; and the town of
Manchester, which contains
upwards of sixty thousand souls, is not admitted to
send any. Is there any principle in these things?”
The framers of the American Constitution announced
that seats in the House of Representatives
would be based on population, which would be determined
every ten years. Nothing, it seemed, could
be simpler.

The constitution provides, “The actual
Enumeration shall be made within
three Years after the first Meeting of the
Congress . . . in such Manner as they shall by Law
direct,” and that “the Number of Representatives
shall not exceed one for every thirty Thousand.”
In 1791, after the census of 1790 put the national
population at 3,929,214, Congress began drafting a bill to apportion the House. The first version—
“An Act for an Apportionment of Representatives
among the Several States According to the First
Enumeration”—passed the Senate in February 1792
and the House in March. It was presented March 26
to President George Washington for his signature.

The president found the bill mathematically
and politically troublesome. Early in April 1792,
Washington shared his concerns with his inner
circle—Secretary of the Treasury Alexander Hamilton,
Secretary of War Henry Knox, Attorney General
Edmund Randolph, and Secretary of State Thomas
Jefferson—and asked for their opinions. Hamilton
and Knox encouraged the president to sign, but Randolph
and Jefferson counselled
a veto.

The skeptics won. April
5, President Washington
wrote from Philadelphia:

I have maturely considered
the Act . . . and
I return it . . . with the
following objections.

First—The Constitution
has prescribed that
representatives shall be
apportioned among the
several States according
to their respective
numbers: and there is
no one proportion or divisor
which, applied to
the respective numbers
of the States will yield
the number and allotment of representatives
proposed by the Bill.

Second—The Constitution has also provided
that the number of Representatives shall
not exceed one for every thirty thousand; which
restriction is, by the context, and by fair and
obvious construction, to be applied to the separate
and respective numbers of the States: and
the bill has allotted to eight of the States, more
than one for thirty thousand.

It was the first time a president rejected any law
passed by Congress, and one of only two Washington
vetoes—an indication of the seriousness with which
he approached the subject.

Washington’s insistence on “one proportion or divisor” is a mathematical expression of the one person,
one vote, principle: no one’s vote should count
for more than anyone else’s. The Congress’s bill,
Washington said, did not meet that standard. Georgia
and Delaware, for instance, each had two representatives,
but their populations differed. Georgia
had one seat for every 35,418 residents, and Delaware
one for every 27,770. A vote in Delaware was,
in effect, worth almost 28 percent more than a vote
in Georgia. The inequity bothered Washington, as
did the fact that eight of the thirteen states violated
a Constitutional mandate by having more than one
representative for every thirty thousand residents.

A new version of the bill followed days after
the veto, based largely on Jefferson’s calculations—
Jefferson, like Benjamin Franklin, had a serious
interest in mathematics, and was among the first
to consider apportionment carefully. The new bill
satisfied Washington and his cabinet, and was soon
signed. But the question still hasn’t been settled to
everyone’s satisfaction.

At first glance, a formula for apportionment
seems simple and intuitive, the sort of
thing that can be jotted on a scrap of paper
in minutes. The size of the House’s membership is
arbitrary; although it has been fixed at 435 since
1913, that provision was passed into law only in
1929 and can be changed by Congress at any time.
But the number of seats can always be known in
advance, and the population of each state will be
known after each census. From these pieces of information
it should be easy, even for the math phobic,
to calculate each state’s apportionment.

Take the population of the United States and
divide it by the number of representatives: that
number, which mathematicians call the ratio, will
give the average number of citizens in each representative’s
district. Divide each state’s population by
that ratio to get the number of representatives that
state is entitled to: this number is the quota. So, for
instance, the United States’ population today is just
over 300 million, and the House seats 435 representatives;
each representative therefore serves approximately
690,000 people. Now consider a state with a
population of 2,070,000: a simple division, 2,070,000
÷ 690,000, gives a quota of 3.0, which means the
state gets three representatives, each representing
a district of 690,000 residents.

So far, so good. But this method leads to frustrations,
because the results are almost never whole
numbers. Imagine three states: state A has a population
of 2,060,000; state B has 2,080,000; and state
C has 2,750,000. Dividing each figure by 690,000
gives 2.99 for A, 3.01 for B, and 3.99 for C. Since
there’s no such thing as a fractional representative,
how many should be assigned to each? How, in other
words, should we deal with the remainders, the fractions
left after the division?

Mathematicians distinguish the exact quota—
the result of that division, which may contain
fractions—from the quota, a whole number with
no fractions. But what’s the fairest way to go from
the fractional number to the whole number? We
could round everything up: state A, since its exact
quota of 2.99 is more than 2.0 but less than 3.0,
would get three representatives; states B and C,
since their exact quotas of 3.01 and 3.99 are more
than 3.0 but less than 4.0, would get four each. Or
we could round down, ignoring the remainders: this
would give state A, with its exact quota of 2.99, two
representatives; states B and C, with exact quotas
more than 3.0 but less than 4.0, would get three
each. This was the policy suggested by Jefferson in
1792—known to mathematicians as “the method
of greatest divisors”—and was used in the United
States through the 1830s.

But the census of 1830 produced the inevitable
difficulty: Daniel Webster pointed out that the exact
quotient for Massachusetts would be 12.99, and
rounding it down to 12 seemed grossly unfair. To
avoid this problem, he proposed rounding small
fractions down and large fractions up: in our imaginary
scenario, this method would give states A and
B three representatives each, and state C four. But
distinguishing “small fractions” from “large fractions”
is not simple. Mathematicians can use such
approaches as “the method of greater fractions,” “the
method of geometric fractions,” or “the method of
harmonic fractions,” but no two methods produce the
same result. And whatever the results, some state is
bound to feel cheated.

To make matters more baffling, when we complete
the apportionment and total the representatives
from each state, none of these methods is likely
to yield a total of 435. Rounding errors, compounded
across all the states, will probably skew the figure.
Some proposed systems—including a so-called Vinton
method, used, with variations, from 1850 to
1940—therefore assign representatives to states with
remainders not on the basis of a simple rounding up
or down, but by allotting the leftover seats round robin until the proper
number is reached.
The mathematician
I. B. Cohen described
a few methods for
distributing the fractions
fairly:

There are other
possible solutions:
one would
be to give an
extra seat to the
state with the
highest fraction,
then to give the
next extra seat
to the state with
the next highest
fraction,
continuing until
the total number
of representatives
adds up.
. . . An alternative
would be
to use a similar
procedure, but
start by giving
an extra seat to
the most populous
state, rather than to the state with the
highest fraction.

And so on. But it turns out that systems like
these, though they seem fairer than picking an arbitrary
policy for rounding up or down, can lead to the
“Alabama paradox.” In 1881, as Congress debated
the preferred size of the House, it was discovered
that, in a House of 299 members, Alabama would
have eight representatives; in a House of 300, they
would have seven. In other words, a state’s delegation
could shrink as the total number of representatives
grew.

If discussions such as these seem abstruse—burdened
as they are with details only a mathematician
could care about—the real-world implications can be
substantial. Much rides on the choice of mathematical
models: a state has reason to care whether it has
two, three, or four votes in the House. As the Yale
Law Journal said,
“The real controversy
concerns selection of
the method of apportionment
closest to
the standard, which
the Constitution must
have implied, of being
fair to every state. To
resolve that controversy,
mathematics
must come to the aid
of law.”

It’s telling that
learned articles on
the subject appear in
journals like Quarterly
Publications of
the American Statistical
Association and
American Mathematical
Monthly: this is a
problem that requires
scrutiny by professional
mathematicians.
But no amount
of arithmetical ingenuity
can ever solve
it once and for all. In
1982, two mathematicians
proved that
every system of apportionment necessarily falls prey
to some paradox.

Since Washington vetoed the first apportionment
bill in 1792, the population of the
United States has multiplied nearly a hundredfold,
and has more than tripled since the size of
the House was set at 435. There’s no longer any danger
of having a representative for fewer than 30,000
residents; in the current apportionment, based on
the census of 2000, each House member represents
more than twenty times the lower limit specified by
the Constitution.

Critics have argued that the size of the House
should be increased: as the number of representatives
grows larger and the districts grow smaller, the
districts become less susceptible to gerrymandering,
the disparities in the Electoral College become less
pronounced, and the difficulties of rounding apportioned seats become
less acute. It’s less
painful to round 100.9
down to 100 than it
is to round 1.9 down
to 1. But such proposals
have attracted
little support, and it
may be impossible to
return to the kinds
of numbers conceived
by the founders. Now
that the country’s
population has passed
300 million, allotting
one representative
to every thirty thousand
people after the
census of 2010 would
mean a House of more
than ten thousand
representatives. It’s
hard to imagine how
a government of that
size could function, to
say nothing of finding
a building large
enough to seat them.

It may be unusual
for articles
on Constitutional
questions to begin, as Michael Balinski and H. Peyton
Young’s “Jefferson Method of Apportionment”
does, with “Let p = (p1, . . . ps) be the populations
of s states, pi > 0 for all i, and h ≥ 0 be the number
of seats. . . .” But this is a case where fundamental
questions of justice require the consultation of
experts, and where a sense of fairness depends on
elaborate calculations.

Yet the mathematicians will never be able to
solve the matter to everyone’s satisfaction, which
means the constitutional question is destined to
last. No system can guarantee the “one proportion
or divisor” that Washington wanted: some votes will
always count more than others. It’s inescapable. As
United States Senator from Massachusetts Daniel
Webster put it in 1830, “Enjoining an absolute relative
equality”—that is, making every vote count for
the same amount—“would be demanding an impossibility.”
He summarizes
two centuries’
efforts to distribute
voting power equitably:
“That which
cannot be done perfectly,
must be done
in a manner as near
perfection as can be.
If exactness cannot,
from the nature of
things, be attained,
then the greatest
practicable approach
to exactness ought to
be made.”

Extra Images

Ellen Knecht and Eomer Knecht interpret the colonial woman and child's role in the census.

Steve Holloway stands as census-taker in this parlor scene.

The 18th-century family had much to gain from their patriarch's vote.

Sylvia Tabb plays the part of an African-American citizen of the colonies.

Jack Lynch, author of Becoming Shakespeare: The Unlikely Afterlife That Turned a Provincial Playwright into the Bard (Walker & Co., 2007) is associate professor of English at Rutgers University. He contributed to the spring 2009 journal “A Principal Source of Dishonor: Indian Policies in Early America.”

Suggestions for further reading:

“Apportionment of the House of Representatives,” Yale Law Journal 58, no. 8 (July 1949): 1360–86.